Engineered granular crystals (EGCs) are closely packed assemblies of particles, contained inside a matrix, interacting through nonlinear, tensionless potentials. They are known as a flexible computational and experimental testbed for nonlinear dynamics  with a large range of applications, including energy absorbing layers , acoustic lenses , and rectifiers . The flexibility of EGCs comes from the large number of tunable parameters in the system. The building blocks of an EGC are macroscopic particles, usually of spherical or cylindrical shape, and by changing the material properties, arrangement and shape of these building blocks, we can modify the dynamic response of the system, and therefore the energy transfer through the system. By applying external forcing to the granular crystal, we can observe periodic, quasi-periodic and chaotic responses.
EGCs belong to a broader category of media that bear a crystalline lattice structure, where the understanding of the underlying linear spectrum is very important for their linear wave transmission properties. However, recently there has been an increased interest in systems that do not possess a linear spectrum and EGCs are a perfect example since their response can be tuned from essentially linear, to weakly and highly nonlinear, depending on the external precompression applied on the crystal.
In this highly nonlinear setting we conduct a bifurcation study of a finite granular crystal with external periodic forcing, using techniques from iterative linear algebra to compute fixed points of the stroboscopic map that correspond to periodic solutions. Combining this method with continuation techniques  we compute one parameter branches of periodic solutions as well as two parameter branches of the bifurcations that occur. Despite the absence of a linear spectrum we identify resonant periodic propagation whereby the crystal responds at multiples of the forcing period and corresponding to local maxima of transmitted force at its right fixed boundary. In addition, we identify anti-resonances in-between resonances corresponding to minima of transmitted force. We identify the complex bifurcation diagram involving period doublings, as well as isolas (of period 3-, 4-, 5- solutions), and study the stability of such states upon frequency and amplitude variations of the external forcing. Furthermore, computing the unstable manifolds of saddle fixed points we are able to determine parameter regions with multiple attractors. This study is a stepping stone in understanding how the construction of an EGC affects the energy transfer through the system, which will allow us subsequently to design crystals that transmit/block signals with specific frequency ranges.
 Nesterenko V. F., Dynamics of Heterogeneous Materials (Springer-Verlag, New York) 2001.
 C. Daraio, V. F. Nesterenko, E. Herbold, and S. Jin, “Energy Trapping and Shock Disintegration in a Composite Granular Medium," Physical Review Letters, vol. 96, p. 058002, Feb. 2006.
 A. Spadoni and C. Daraio, “Generation and control of sound bullets with a nonlinear acoustic lens,” Proceedings of the National Academy of Sciences, vol. 107, pp. 7230–7234, Apr. 2010.
 N. Boechler, G. Theocharis, and C. Daraio, “Bifurcation-based acoustic switching and rectification," Nature materials, vol. 10, pp. 665-668, Sept. 2011.
 E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (I): Bifurcation in finite dimensions,” International Journal of Bifurcation and Chaos, vol. 1, no. 03, pp. 493–520, 1991.
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